Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3

Percentage Accurate: 93.5% → 99.6%

Time: 15.3s

Alternatives: 10

Speedup: 1.0×

• 24.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) * (3.0d0 - x)) / (y * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(y * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) * (3.0 - x)) / (y * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) * (3.0d0 - x)) / (y * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(y * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) * (3.0 - x)) / (y * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (- 1.0 x) (/ (- 3.0 x) (* 3.0 y))))

C

double code(double x, double y) {
	return (1.0 - x) * ((3.0 - x) / (3.0 * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) * ((3.0d0 - x) / (3.0d0 * y))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) * ((3.0 - x) / (3.0 * y));
}

Python

def code(x, y):
	return (1.0 - x) * ((3.0 - x) / (3.0 * y))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) * Float64(Float64(3.0 - x) / Float64(3.0 * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) * ((3.0 - x) / (3.0 * y));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] * N[(N[(3.0 - x), $MachinePrecision] / N[(3.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
2. Step-by-step derivation

   1. associate-/l* 99.6%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

   2. *-commutative 99.6%

      \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(\left(3 - x\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.3) (not (<= x 1.3)))
   (* -0.3333333333333333 (* (- 3.0 x) (/ x y)))
   (+ (* -1.3333333333333333 (/ x y)) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.3) || !(x <= 1.3)) {
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	} else {
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.3d0)) .or. (.not. (x <= 1.3d0))) then
        tmp = (-0.3333333333333333d0) * ((3.0d0 - x) * (x / y))
    else
        tmp = ((-1.3333333333333333d0) * (x / y)) + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.3) || !(x <= 1.3)) {
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	} else {
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.3) or not (x <= 1.3):
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y))
	else:
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.3) || !(x <= 1.3))
		tmp = Float64(-0.3333333333333333 * Float64(Float64(3.0 - x) * Float64(x / y)));
	else
		tmp = Float64(Float64(-1.3333333333333333 * Float64(x / y)) + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.3) || ~((x <= 1.3)))
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	else
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.3], N[Not[LessEqual[x, 1.3]], $MachinePrecision]], N[(-0.3333333333333333 * N[(N[(3.0 - x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.3333333333333333 * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999998 or 1.30000000000000004 < x

   1. Initial program 92.5%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

         \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

      2. *-commutative 99.7%

         \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 97.7%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   7. Simplified 97.7%

      \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   8. Taylor expanded in y around 0 90.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{x \cdot \left(3 - x\right)}{y}} \]
   9. Step-by-step derivation

      1. associate-*r/ 90.5%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(x \cdot \left(3 - x\right)\right)}{y}} \]

   10. Simplified 90.5%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(x \cdot \left(3 - x\right)\right)}{y}} \]
   11. Step-by-step derivation

       1. associate-/l* 90.3%

          \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{x \cdot \left(3 - x\right)}{y}} \]

       2. *-commutative 90.3%

          \[\leadsto \color{blue}{\frac{x \cdot \left(3 - x\right)}{y} \cdot -0.3333333333333333} \]

       3. *-commutative 90.3%

          \[\leadsto \frac{\color{blue}{\left(3 - x\right) \cdot x}}{y} \cdot -0.3333333333333333 \]

       4. associate-/l* 97.6%

          \[\leadsto \color{blue}{\left(\left(3 - x\right) \cdot \frac{x}{y}\right)} \cdot -0.3333333333333333 \]

   12. Applied egg-rr 97.6%

       \[\leadsto \color{blue}{\left(\left(3 - x\right) \cdot \frac{x}{y}\right) \cdot -0.3333333333333333} \]

   if -2.2999999999999998 < x < 1.30000000000000004

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(\left(3 - x\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.78 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(\left(3 - x\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.78) (not (<= x 1.0)))
   (* -0.3333333333333333 (* (- 3.0 x) (/ x y)))
   (/ 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.78) || !(x <= 1.0)) {
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	} else {
		tmp = 1.0 / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.78d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-0.3333333333333333d0) * ((3.0d0 - x) * (x / y))
    else
        tmp = 1.0d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.78) || !(x <= 1.0)) {
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	} else {
		tmp = 1.0 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.78) or not (x <= 1.0):
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y))
	else:
		tmp = 1.0 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.78) || !(x <= 1.0))
		tmp = Float64(-0.3333333333333333 * Float64(Float64(3.0 - x) * Float64(x / y)));
	else
		tmp = Float64(1.0 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.78) || ~((x <= 1.0)))
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	else
		tmp = 1.0 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.78], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(-0.3333333333333333 * N[(N[(3.0 - x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.78000000000000003 or 1 < x

   1. Initial program 92.5%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

         \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

      2. *-commutative 99.7%

         \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 97.7%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   7. Simplified 97.7%

      \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   8. Taylor expanded in y around 0 90.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{x \cdot \left(3 - x\right)}{y}} \]
   9. Step-by-step derivation

      1. associate-*r/ 90.5%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(x \cdot \left(3 - x\right)\right)}{y}} \]

   10. Simplified 90.5%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(x \cdot \left(3 - x\right)\right)}{y}} \]
   11. Step-by-step derivation

       1. associate-/l* 90.3%

          \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{x \cdot \left(3 - x\right)}{y}} \]

       2. *-commutative 90.3%

          \[\leadsto \color{blue}{\frac{x \cdot \left(3 - x\right)}{y} \cdot -0.3333333333333333} \]

       3. *-commutative 90.3%

          \[\leadsto \frac{\color{blue}{\left(3 - x\right) \cdot x}}{y} \cdot -0.3333333333333333 \]

       4. associate-/l* 97.6%

          \[\leadsto \color{blue}{\left(\left(3 - x\right) \cdot \frac{x}{y}\right)} \cdot -0.3333333333333333 \]

   12. Applied egg-rr 97.6%

       \[\leadsto \color{blue}{\left(\left(3 - x\right) \cdot \frac{x}{y}\right) \cdot -0.3333333333333333} \]

   if -0.78000000000000003 < x < 1

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{\frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.78 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(\left(3 - x\right) \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;x \cdot \frac{x - 3}{3 \cdot y}\\ \mathbf{elif}\;x \leq 0.405:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{3 \cdot y}{x + 3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.3)
   (* x (/ (- x 3.0) (* 3.0 y)))
   (if (<= x 0.405)
     (+ (* -1.3333333333333333 (/ x y)) (/ 1.0 y))
     (/ x (/ (* 3.0 y) (+ x 3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.3) {
		tmp = x * ((x - 3.0) / (3.0 * y));
	} else if (x <= 0.405) {
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	} else {
		tmp = x / ((3.0 * y) / (x + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.3d0)) then
        tmp = x * ((x - 3.0d0) / (3.0d0 * y))
    else if (x <= 0.405d0) then
        tmp = ((-1.3333333333333333d0) * (x / y)) + (1.0d0 / y)
    else
        tmp = x / ((3.0d0 * y) / (x + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.3) {
		tmp = x * ((x - 3.0) / (3.0 * y));
	} else if (x <= 0.405) {
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	} else {
		tmp = x / ((3.0 * y) / (x + 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.3:
		tmp = x * ((x - 3.0) / (3.0 * y))
	elif x <= 0.405:
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y)
	else:
		tmp = x / ((3.0 * y) / (x + 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.3)
		tmp = Float64(x * Float64(Float64(x - 3.0) / Float64(3.0 * y)));
	elseif (x <= 0.405)
		tmp = Float64(Float64(-1.3333333333333333 * Float64(x / y)) + Float64(1.0 / y));
	else
		tmp = Float64(x / Float64(Float64(3.0 * y) / Float64(x + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.3)
		tmp = x * ((x - 3.0) / (3.0 * y));
	elseif (x <= 0.405)
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	else
		tmp = x / ((3.0 * y) / (x + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.3], N[(x * N[(N[(x - 3.0), $MachinePrecision] / N[(3.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.405], N[(N[(-1.3333333333333333 * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(3.0 * y), $MachinePrecision] / N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2999999999999998

   1. Initial program 87.8%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

      2. *-commutative 99.8%

         \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.6%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 97.6%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   7. Simplified 97.6%

      \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   if -2.2999999999999998 < x < 0.40500000000000003

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}} \]

   if 0.40500000000000003 < x

   1. Initial program 96.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Step-by-step derivation

      1. associate-/l* 99.6%

         \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

      2. *-commutative 99.6%

         \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 97.7%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   7. Simplified 97.7%

      \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   8. Step-by-step derivation

      1. clear-num 97.8%

         \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{3 \cdot y}{3 - x}}} \]

      2. un-div-inv 97.9%

         \[\leadsto \color{blue}{\frac{-x}{\frac{3 \cdot y}{3 - x}}} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{\frac{3 \cdot y}{3 - x}} \]

      4. sqrt-unprod 0.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{\frac{3 \cdot y}{3 - x}} \]

      5. sqr-neg 0.7%

         \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x}}}{\frac{3 \cdot y}{3 - x}} \]

      6. sqrt-unprod 0.7%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\frac{3 \cdot y}{3 - x}} \]

      7. add-sqr-sqrt 0.7%

         \[\leadsto \frac{\color{blue}{x}}{\frac{3 \cdot y}{3 - x}} \]

      8. *-commutative 0.7%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot 3}}{3 - x}} \]

      9. sub-neg 0.7%

         \[\leadsto \frac{x}{\frac{y \cdot 3}{\color{blue}{3 + \left(-x\right)}}} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{3 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}} \]

      11. sqrt-unprod 94.6%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{3 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}} \]

      12. sqr-neg 94.6%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{3 + \sqrt{\color{blue}{x \cdot x}}}} \]

      13. sqrt-unprod 97.5%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{3 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

      14. add-sqr-sqrt 97.7%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{3 + \color{blue}{x}}} \]

      15. +-commutative 97.7%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{\color{blue}{x + 3}}} \]

   9. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot 3}{x + 3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;x \cdot \frac{x - 3}{3 \cdot y}\\ \mathbf{elif}\;x \leq 0.405:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{3 \cdot y}{x + 3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\left(3 - x\right) \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 0.405:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{3 \cdot y}{x + 3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.3)
   (* -0.3333333333333333 (* (- 3.0 x) (/ x y)))
   (if (<= x 0.405)
     (+ (* -1.3333333333333333 (/ x y)) (/ 1.0 y))
     (/ x (/ (* 3.0 y) (+ x 3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.3) {
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	} else if (x <= 0.405) {
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	} else {
		tmp = x / ((3.0 * y) / (x + 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.3d0)) then
        tmp = (-0.3333333333333333d0) * ((3.0d0 - x) * (x / y))
    else if (x <= 0.405d0) then
        tmp = ((-1.3333333333333333d0) * (x / y)) + (1.0d0 / y)
    else
        tmp = x / ((3.0d0 * y) / (x + 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.3) {
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	} else if (x <= 0.405) {
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	} else {
		tmp = x / ((3.0 * y) / (x + 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.3:
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y))
	elif x <= 0.405:
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y)
	else:
		tmp = x / ((3.0 * y) / (x + 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.3)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(3.0 - x) * Float64(x / y)));
	elseif (x <= 0.405)
		tmp = Float64(Float64(-1.3333333333333333 * Float64(x / y)) + Float64(1.0 / y));
	else
		tmp = Float64(x / Float64(Float64(3.0 * y) / Float64(x + 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.3)
		tmp = -0.3333333333333333 * ((3.0 - x) * (x / y));
	elseif (x <= 0.405)
		tmp = (-1.3333333333333333 * (x / y)) + (1.0 / y);
	else
		tmp = x / ((3.0 * y) / (x + 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.3], N[(-0.3333333333333333 * N[(N[(3.0 - x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.405], N[(N[(-1.3333333333333333 * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(3.0 * y), $MachinePrecision] / N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2999999999999998

   1. Initial program 87.8%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

      2. *-commutative 99.8%

         \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.6%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 97.6%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   7. Simplified 97.6%

      \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   8. Taylor expanded in y around 0 85.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{x \cdot \left(3 - x\right)}{y}} \]
   9. Step-by-step derivation

      1. associate-*r/ 85.6%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(x \cdot \left(3 - x\right)\right)}{y}} \]

   10. Simplified 85.6%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(x \cdot \left(3 - x\right)\right)}{y}} \]
   11. Step-by-step derivation

       1. associate-/l* 85.6%

          \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{x \cdot \left(3 - x\right)}{y}} \]

       2. *-commutative 85.6%

          \[\leadsto \color{blue}{\frac{x \cdot \left(3 - x\right)}{y} \cdot -0.3333333333333333} \]

       3. *-commutative 85.6%

          \[\leadsto \frac{\color{blue}{\left(3 - x\right) \cdot x}}{y} \cdot -0.3333333333333333 \]

       4. associate-/l* 97.5%

          \[\leadsto \color{blue}{\left(\left(3 - x\right) \cdot \frac{x}{y}\right)} \cdot -0.3333333333333333 \]

   12. Applied egg-rr 97.5%

       \[\leadsto \color{blue}{\left(\left(3 - x\right) \cdot \frac{x}{y}\right) \cdot -0.3333333333333333} \]

   if -2.2999999999999998 < x < 0.40500000000000003

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}} \]

   if 0.40500000000000003 < x

   1. Initial program 96.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Step-by-step derivation

      1. associate-/l* 99.6%

         \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

      2. *-commutative 99.6%

         \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 97.7%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   7. Simplified 97.7%

      \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   8. Step-by-step derivation

      1. clear-num 97.8%

         \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{1}{\frac{3 \cdot y}{3 - x}}} \]

      2. un-div-inv 97.9%

         \[\leadsto \color{blue}{\frac{-x}{\frac{3 \cdot y}{3 - x}}} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{\frac{3 \cdot y}{3 - x}} \]

      4. sqrt-unprod 0.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{\frac{3 \cdot y}{3 - x}} \]

      5. sqr-neg 0.7%

         \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x}}}{\frac{3 \cdot y}{3 - x}} \]

      6. sqrt-unprod 0.7%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\frac{3 \cdot y}{3 - x}} \]

      7. add-sqr-sqrt 0.7%

         \[\leadsto \frac{\color{blue}{x}}{\frac{3 \cdot y}{3 - x}} \]

      8. *-commutative 0.7%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot 3}}{3 - x}} \]

      9. sub-neg 0.7%

         \[\leadsto \frac{x}{\frac{y \cdot 3}{\color{blue}{3 + \left(-x\right)}}} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{3 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}} \]

      11. sqrt-unprod 94.6%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{3 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}} \]

      12. sqr-neg 94.6%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{3 + \sqrt{\color{blue}{x \cdot x}}}} \]

      13. sqrt-unprod 97.5%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{3 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}} \]

      14. add-sqr-sqrt 97.7%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{3 + \color{blue}{x}}} \]

      15. +-commutative 97.7%

          \[\leadsto \frac{x}{\frac{y \cdot 3}{\color{blue}{x + 3}}} \]

   9. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{y \cdot 3}{x + 3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\left(3 - x\right) \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 0.405:\\ \;\;\;\;-1.3333333333333333 \cdot \frac{x}{y} + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{3 \cdot y}{x + 3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-\frac{-0.3333333333333333}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.75) (not (<= x 0.58)))
   (* x (* x (- (/ -0.3333333333333333 y))))
   (/ 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.75) || !(x <= 0.58)) {
		tmp = x * (x * -(-0.3333333333333333 / y));
	} else {
		tmp = 1.0 / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.75d0)) .or. (.not. (x <= 0.58d0))) then
        tmp = x * (x * -((-0.3333333333333333d0) / y))
    else
        tmp = 1.0d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.75) || !(x <= 0.58)) {
		tmp = x * (x * -(-0.3333333333333333 / y));
	} else {
		tmp = 1.0 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.75) or not (x <= 0.58):
		tmp = x * (x * -(-0.3333333333333333 / y))
	else:
		tmp = 1.0 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.75) || !(x <= 0.58))
		tmp = Float64(x * Float64(x * Float64(-Float64(-0.3333333333333333 / y))));
	else
		tmp = Float64(1.0 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.75) || ~((x <= 0.58)))
		tmp = x * (x * -(-0.3333333333333333 / y));
	else
		tmp = 1.0 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.75], N[Not[LessEqual[x, 0.58]], $MachinePrecision]], N[(x * N[(x * (-N[(-0.3333333333333333 / y), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(1.0 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75 or 0.57999999999999996 < x

   1. Initial program 92.5%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

         \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

      2. *-commutative 99.7%

         \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.5%

      \[\leadsto \left(1 - x\right) \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{x}{y}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 97.6%

         \[\leadsto \left(1 - x\right) \cdot \color{blue}{\frac{-0.3333333333333333 \cdot x}{y}} \]

      2. associate-*l/ 97.5%

         \[\leadsto \left(1 - x\right) \cdot \color{blue}{\left(\frac{-0.3333333333333333}{y} \cdot x\right)} \]

      3. *-commutative 97.5%

         \[\leadsto \left(1 - x\right) \cdot \color{blue}{\left(x \cdot \frac{-0.3333333333333333}{y}\right)} \]

   7. Simplified 97.5%

      \[\leadsto \left(1 - x\right) \cdot \color{blue}{\left(x \cdot \frac{-0.3333333333333333}{y}\right)} \]

   8. Taylor expanded in x around inf 97.5%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \left(x \cdot \frac{-0.3333333333333333}{y}\right) \]
   9. Step-by-step derivation

      1. neg-mul-1 97.7%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   10. Simplified 97.5%

       \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(x \cdot \frac{-0.3333333333333333}{y}\right) \]

   if -1.75 < x < 0.57999999999999996

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{\frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \lor \neg \left(x \leq 0.58\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(-\frac{-0.3333333333333333}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 0.34:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ (- x) y) (if (<= x 0.34) (/ 1.0 y) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -x / y;
	} else if (x <= 0.34) {
		tmp = 1.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -x / y
    else if (x <= 0.34d0) then
        tmp = 1.0d0 / y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -x / y;
	} else if (x <= 0.34) {
		tmp = 1.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -x / y
	elif x <= 0.34:
		tmp = 1.0 / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(-x) / y);
	elseif (x <= 0.34)
		tmp = Float64(1.0 / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -x / y;
	elseif (x <= 0.34)
		tmp = 1.0 / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[((-x) / y), $MachinePrecision], If[LessEqual[x, 0.34], N[(1.0 / y), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 87.8%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

      2. *-commutative 99.8%

         \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.6%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 97.6%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   7. Simplified 97.6%

      \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   8. Taylor expanded in x around 0 32.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. neg-mul-1 32.7%

         \[\leadsto \color{blue}{-\frac{x}{y}} \]

      2. distribute-neg-frac2 32.7%

         \[\leadsto \color{blue}{\frac{x}{-y}} \]

   10. Simplified 32.7%

       \[\leadsto \color{blue}{\frac{x}{-y}} \]

   if -1 < x < 0.340000000000000024

   1. Initial program 99.6%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{\frac{1}{y}} \]

   if 0.340000000000000024 < x

   1. Initial program 96.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Step-by-step derivation

      1. associate-/l* 99.6%

         \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

      2. *-commutative 99.6%

         \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 97.7%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   7. Simplified 97.7%

      \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   8. Step-by-step derivation

      1. associate-*r/ 94.8%

         \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \left(3 - x\right)}{3 \cdot y}} \]

      2. associate-/r* 94.8%

         \[\leadsto \color{blue}{\frac{\frac{\left(-x\right) \cdot \left(3 - x\right)}{3}}{y}} \]

      3. sub-neg 94.8%

         \[\leadsto \frac{\frac{\left(-x\right) \cdot \color{blue}{\left(3 + \left(-x\right)\right)}}{3}}{y} \]

      4. distribute-rgt-in 94.8%

         \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(-x\right) + \left(-x\right) \cdot \left(-x\right)}}{3}}{y} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\frac{3 \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} + \left(-x\right) \cdot \left(-x\right)}{3}}{y} \]

      6. sqrt-unprod 94.6%

         \[\leadsto \frac{\frac{3 \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + \left(-x\right) \cdot \left(-x\right)}{3}}{y} \]

      7. sqr-neg 94.6%

         \[\leadsto \frac{\frac{3 \cdot \sqrt{\color{blue}{x \cdot x}} + \left(-x\right) \cdot \left(-x\right)}{3}}{y} \]

      8. sqrt-unprod 94.6%

         \[\leadsto \frac{\frac{3 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + \left(-x\right) \cdot \left(-x\right)}{3}}{y} \]

      9. add-sqr-sqrt 94.6%

         \[\leadsto \frac{\frac{3 \cdot \color{blue}{x} + \left(-x\right) \cdot \left(-x\right)}{3}}{y} \]

      10. sqr-neg 94.6%

          \[\leadsto \frac{\frac{3 \cdot x + \color{blue}{x \cdot x}}{3}}{y} \]

      11. +-commutative 94.6%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot x + 3 \cdot x}}{3}}{y} \]

      12. distribute-rgt-out 94.6%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x + 3\right)}}{3}}{y} \]

   9. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot \left(x + 3\right)}{3}}{y}} \]

   10. Taylor expanded in x around 0 25.6%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;x \leq 0.34:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) \cdot \left(\left(x + -3\right) \cdot \frac{-0.3333333333333333}{y}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (- 1.0 x) (* (+ x -3.0) (/ -0.3333333333333333 y))))

C

double code(double x, double y) {
	return (1.0 - x) * ((x + -3.0) * (-0.3333333333333333 / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) * ((x + (-3.0d0)) * ((-0.3333333333333333d0) / y))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) * ((x + -3.0) * (-0.3333333333333333 / y));
}

Python

def code(x, y):
	return (1.0 - x) * ((x + -3.0) * (-0.3333333333333333 / y))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) * Float64(Float64(x + -3.0) * Float64(-0.3333333333333333 / y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) * ((x + -3.0) * (-0.3333333333333333 / y));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] * N[(N[(x + -3.0), $MachinePrecision] * N[(-0.3333333333333333 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
2. Step-by-step derivation

   1. associate-/l* 99.6%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

   2. *-rgt-identity 99.6%

      \[\leadsto \left(1 - x\right) \cdot \frac{\color{blue}{\left(3 - x\right) \cdot 1}}{y \cdot 3} \]

   3. remove-double-neg 99.6%

      \[\leadsto \left(1 - x\right) \cdot \frac{\left(3 - x\right) \cdot 1}{\color{blue}{-\left(-y \cdot 3\right)}} \]

   4. distribute-lft-neg-out 99.6%

      \[\leadsto \left(1 - x\right) \cdot \frac{\left(3 - x\right) \cdot 1}{-\color{blue}{\left(-y\right) \cdot 3}} \]

   5. neg-mul-1 99.6%

      \[\leadsto \left(1 - x\right) \cdot \frac{\left(3 - x\right) \cdot 1}{\color{blue}{-1 \cdot \left(\left(-y\right) \cdot 3\right)}} \]

   6. times-frac 99.5%

      \[\leadsto \left(1 - x\right) \cdot \color{blue}{\left(\frac{3 - x}{-1} \cdot \frac{1}{\left(-y\right) \cdot 3}\right)} \]

   7. *-rgt-identity 99.5%

      \[\leadsto \left(1 - x\right) \cdot \left(\frac{\color{blue}{\left(3 - x\right) \cdot 1}}{-1} \cdot \frac{1}{\left(-y\right) \cdot 3}\right) \]

   8. associate-/l* 99.5%

      \[\leadsto \left(1 - x\right) \cdot \left(\color{blue}{\left(\left(3 - x\right) \cdot \frac{1}{-1}\right)} \cdot \frac{1}{\left(-y\right) \cdot 3}\right) \]

   9. metadata-eval 99.5%

      \[\leadsto \left(1 - x\right) \cdot \left(\left(\left(3 - x\right) \cdot \color{blue}{-1}\right) \cdot \frac{1}{\left(-y\right) \cdot 3}\right) \]

   10. *-commutative 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\color{blue}{\left(-1 \cdot \left(3 - x\right)\right)} \cdot \frac{1}{\left(-y\right) \cdot 3}\right) \]

   11. sub-neg 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(-1 \cdot \color{blue}{\left(3 + \left(-x\right)\right)}\right) \cdot \frac{1}{\left(-y\right) \cdot 3}\right) \]

   12. +-commutative 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(-1 \cdot \color{blue}{\left(\left(-x\right) + 3\right)}\right) \cdot \frac{1}{\left(-y\right) \cdot 3}\right) \]

   13. distribute-lft-in 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\color{blue}{\left(-1 \cdot \left(-x\right) + -1 \cdot 3\right)} \cdot \frac{1}{\left(-y\right) \cdot 3}\right) \]

   14. neg-mul-1 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(\color{blue}{\left(-\left(-x\right)\right)} + -1 \cdot 3\right) \cdot \frac{1}{\left(-y\right) \cdot 3}\right) \]

   15. remove-double-neg 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(\color{blue}{x} + -1 \cdot 3\right) \cdot \frac{1}{\left(-y\right) \cdot 3}\right) \]

   16. metadata-eval 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(x + \color{blue}{-3}\right) \cdot \frac{1}{\left(-y\right) \cdot 3}\right) \]

   17. distribute-lft-neg-out 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(x + -3\right) \cdot \frac{1}{\color{blue}{-y \cdot 3}}\right) \]

   18. *-commutative 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(x + -3\right) \cdot \frac{1}{-\color{blue}{3 \cdot y}}\right) \]

   19. distribute-lft-neg-in 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(x + -3\right) \cdot \frac{1}{\color{blue}{\left(-3\right) \cdot y}}\right) \]

   20. associate-/r* 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(x + -3\right) \cdot \color{blue}{\frac{\frac{1}{-3}}{y}}\right) \]

   21. metadata-eval 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(x + -3\right) \cdot \frac{\frac{1}{\color{blue}{-3}}}{y}\right) \]

   22. metadata-eval 99.5%

       \[\leadsto \left(1 - x\right) \cdot \left(\left(x + -3\right) \cdot \frac{\color{blue}{-0.3333333333333333}}{y}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\left(x + -3\right) \cdot \frac{-0.3333333333333333}{y}\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 9: 57.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.34:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 0.34) (/ 1.0 y) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.34) {
		tmp = 1.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.34d0) then
        tmp = 1.0d0 / y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.34) {
		tmp = 1.0 / y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.34:
		tmp = 1.0 / y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.34)
		tmp = Float64(1.0 / y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.34)
		tmp = 1.0 / y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.34], N[(1.0 / y), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.340000000000000024

   1. Initial program 96.4%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{\frac{1}{y}} \]

   if 0.340000000000000024 < x

   1. Initial program 96.7%

      \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
   2. Step-by-step derivation

      1. associate-/l* 99.6%

         \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{y \cdot 3}} \]

      2. *-commutative 99.6%

         \[\leadsto \left(1 - x\right) \cdot \frac{3 - x}{\color{blue}{3 \cdot y}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 - x\right) \cdot \frac{3 - x}{3 \cdot y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 97.7%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]

   7. Simplified 97.7%

      \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{3 - x}{3 \cdot y} \]
   8. Step-by-step derivation

      1. associate-*r/ 94.8%

         \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \left(3 - x\right)}{3 \cdot y}} \]

      2. associate-/r* 94.8%

         \[\leadsto \color{blue}{\frac{\frac{\left(-x\right) \cdot \left(3 - x\right)}{3}}{y}} \]

      3. sub-neg 94.8%

         \[\leadsto \frac{\frac{\left(-x\right) \cdot \color{blue}{\left(3 + \left(-x\right)\right)}}{3}}{y} \]

      4. distribute-rgt-in 94.8%

         \[\leadsto \frac{\frac{\color{blue}{3 \cdot \left(-x\right) + \left(-x\right) \cdot \left(-x\right)}}{3}}{y} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\frac{3 \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} + \left(-x\right) \cdot \left(-x\right)}{3}}{y} \]

      6. sqrt-unprod 94.6%

         \[\leadsto \frac{\frac{3 \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + \left(-x\right) \cdot \left(-x\right)}{3}}{y} \]

      7. sqr-neg 94.6%

         \[\leadsto \frac{\frac{3 \cdot \sqrt{\color{blue}{x \cdot x}} + \left(-x\right) \cdot \left(-x\right)}{3}}{y} \]

      8. sqrt-unprod 94.6%

         \[\leadsto \frac{\frac{3 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + \left(-x\right) \cdot \left(-x\right)}{3}}{y} \]

      9. add-sqr-sqrt 94.6%

         \[\leadsto \frac{\frac{3 \cdot \color{blue}{x} + \left(-x\right) \cdot \left(-x\right)}{3}}{y} \]

      10. sqr-neg 94.6%

          \[\leadsto \frac{\frac{3 \cdot x + \color{blue}{x \cdot x}}{3}}{y} \]

      11. +-commutative 94.6%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot x + 3 \cdot x}}{3}}{y} \]

      12. distribute-rgt-out 94.6%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x + 3\right)}}{3}}{y} \]

   9. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot \left(x + 3\right)}{3}}{y}} \]

   10. Taylor expanded in x around 0 25.6%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 y))

C

double code(double x, double y) {
	return 1.0 / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / y
end function

Java

public static double code(double x, double y) {
	return 1.0 / y;
}

Python

def code(x, y):
	return 1.0 / y

Julia

function code(x, y)
	return Float64(1.0 / y)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / y;
end

Wolfram

code[x_, y_] := N[(1.0 / y), $MachinePrecision]

Derivation

1. Initial program 96.5%

   \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 56.9%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
4. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{y} \cdot \frac{3 - x}{3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0)))

C

double code(double x, double y) {
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((1.0d0 - x) / y) * ((3.0d0 - x) / 3.0d0)
end function

Java

public static double code(double x, double y) {
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0);
}

Python

def code(x, y):
	return ((1.0 - x) / y) * ((3.0 - x) / 3.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) / y) * Float64(Float64(3.0 - x) / 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((1.0 - x) / y) * ((3.0 - x) / 3.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] * N[(N[(3.0 - x), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024111 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0))

  (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))